Volume 10, Issue 2
Using Parity to Accelerate the Computation of the Zeros of Truncated Legendre and Gegenbauer Polynomial Series and Gaussian Quadrature

East Asian J. Appl. Math., 10 (2020), pp. 217-242.

Published online: 2020-04

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• Abstract

Any function $u(x)$ can be decomposed into its parts that are symmetric and antisymmetric with respect to the origin. The zeros, maxima and minima of a truncated spectral series of degree $N$ can always be computed as the eigenvalues of the sparse $N$-dimensional companion matrix whose elements are trivial functions of the coefﬁcients of the spectral series. Here, we show that the matrix dimension can be halved if the series has deﬁnite parity. A series of Legendre and Gegenbauer polynomials has even parity if only even degree coefﬁcients are nonzero and odd parity if the sum includes odd degrees only. We give the elements of the parity-exploiting companion matrices explicitly. We also give the coefﬁcients of parity-exploiting recurrences for computing the orthogonal polynomials of even degree only or odd degree only without the wasteful computation of all polynomials of the opposite parity. For an $N$-point Gaussian quadrature, the quadrature points are the eigenvalues of a symmetric tridiagonal matrix of dimension $N$ ("Jacobi matrix"). We give the explicit elements of symmetric tridiagonal matrices of dimension $N$/2 that do the same job.

• Keywords

Gaussian quadrature, Legendre polynomials, Gegenbauer polynomials, parity, rootﬁnding.

65D32, 65H04, 365H05

jpboyd@umich.edu (John P. Boyd)

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@Article{EAJAM-10-217, author = {P. Boyd , John }, title = {Using Parity to Accelerate the Computation of the Zeros of Truncated Legendre and Gegenbauer Polynomial Series and Gaussian Quadrature}, journal = {East Asian Journal on Applied Mathematics}, year = {2020}, volume = {10}, number = {2}, pages = {217--242}, abstract = {

Any function $u(x)$ can be decomposed into its parts that are symmetric and antisymmetric with respect to the origin. The zeros, maxima and minima of a truncated spectral series of degree $N$ can always be computed as the eigenvalues of the sparse $N$-dimensional companion matrix whose elements are trivial functions of the coefﬁcients of the spectral series. Here, we show that the matrix dimension can be halved if the series has deﬁnite parity. A series of Legendre and Gegenbauer polynomials has even parity if only even degree coefﬁcients are nonzero and odd parity if the sum includes odd degrees only. We give the elements of the parity-exploiting companion matrices explicitly. We also give the coefﬁcients of parity-exploiting recurrences for computing the orthogonal polynomials of even degree only or odd degree only without the wasteful computation of all polynomials of the opposite parity. For an $N$-point Gaussian quadrature, the quadrature points are the eigenvalues of a symmetric tridiagonal matrix of dimension $N$ ("Jacobi matrix"). We give the explicit elements of symmetric tridiagonal matrices of dimension $N$/2 that do the same job.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.140819.271119}, url = {http://global-sci.org/intro/article_detail/eajam/16135.html} }
TY - JOUR T1 - Using Parity to Accelerate the Computation of the Zeros of Truncated Legendre and Gegenbauer Polynomial Series and Gaussian Quadrature AU - P. Boyd , John JO - East Asian Journal on Applied Mathematics VL - 2 SP - 217 EP - 242 PY - 2020 DA - 2020/04 SN - 10 DO - http://dor.org/10.4208/eajam.140819.271119 UR - https://global-sci.org/intro/article_detail/eajam/16135.html KW - Gaussian quadrature, Legendre polynomials, Gegenbauer polynomials, parity, rootﬁnding. AB -

Any function $u(x)$ can be decomposed into its parts that are symmetric and antisymmetric with respect to the origin. The zeros, maxima and minima of a truncated spectral series of degree $N$ can always be computed as the eigenvalues of the sparse $N$-dimensional companion matrix whose elements are trivial functions of the coefﬁcients of the spectral series. Here, we show that the matrix dimension can be halved if the series has deﬁnite parity. A series of Legendre and Gegenbauer polynomials has even parity if only even degree coefﬁcients are nonzero and odd parity if the sum includes odd degrees only. We give the elements of the parity-exploiting companion matrices explicitly. We also give the coefﬁcients of parity-exploiting recurrences for computing the orthogonal polynomials of even degree only or odd degree only without the wasteful computation of all polynomials of the opposite parity. For an $N$-point Gaussian quadrature, the quadrature points are the eigenvalues of a symmetric tridiagonal matrix of dimension $N$ ("Jacobi matrix"). We give the explicit elements of symmetric tridiagonal matrices of dimension $N$/2 that do the same job.

John P. Boyd. (2020). Using Parity to Accelerate the Computation of the Zeros of Truncated Legendre and Gegenbauer Polynomial Series and Gaussian Quadrature. East Asian Journal on Applied Mathematics. 10 (2). 217-242. doi:10.4208/eajam.140819.271119
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